# Questions tagged [bivariate-distributions]

For questions on bivariate distribution, the joint probability distribution of two random variables.

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### Bivariate distribution of angles between two vectors and random vector

Let $Z_1 \in \mathbb{R}^d$ and $Z_2\in\mathbb{R}^d$ be two nonzero vectors. Let $X$ be a random vector distributed uniformly on the hypersphere $\mathbb{S}^{d-1}$. In the case $d=2$, the marginal ...
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### Conditional mean and variance of two Gaussians given one is larger.

Given two Gaussian random variables $Z:=(Z_1,Z_2)\sim N(0,\Sigma)$, where $\Sigma\in\mathbb{R}^{2\times 2}$, is there an explicit expression for the mean and variance of $Z$ conditioned on $Z_1>Z_2$...
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### Posterior distribution of a bivariate normal distribution

I have three continuous random variables $Z, X_1, X_2$ that are related as $$Z = a_1 \times X_1 + a_2 \times X_2 + \varepsilon = \textbf{a'} \textbf{X} + \varepsilon,$$ where $\varepsilon \sim N(0,1)$...
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### Finding Correlation of a Bivariate Function

Here is a bivariate function with the variables $i,j$ is given as: $f(i,j) = Wij(i+j),$ $0<i<1,$ $0<j<1.$ Where $W$ is just a constant. Assume that $M=\min(I,J)$ and $N = \max(I,J).$ ...
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### If there's another variable that controls whether heads outcomes are allowed, what distribution would that coin toss follow?

For a usual coin with probability of heads $P(H)$, the distribution of the numbers of heads after $n$ tosses follows a binomial distribution. The coin can be biased or unbiased. But suppose there's ...
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### Confusing in limits of bivarite distribution and contradiction in related question

This question is from "Mathematical Statistics with application by Wackerly and Mendelhall" $8th$ edition page $233$ ,question $5.5$ : $f(y_1,y_2)=3y_1 , 0 \leq y_2 \leq y_1 \leq1$ and $0$ ,... 41 views

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### Expectation of $|x^{2} - y^{2}|$ when $x$ and $y$ are bivariate normal

I was asked the following question by my professor. Suppose $x$ and $y$ are jointly normal variables. We further suppose the marginal distribution of $x$ and $y$ are $N(0, 1)$ and $N(0, 2)$ ...
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Let $X=(X_1,X_2,X_3)^T$ be a multivariate random variable with the standard Gaussian distribution on $\mathbb{R}^3$. Define the multivariate random variable $Y=(Y_1,Y_2,Y_3)^T$ by $$\begin{pmatrix} ... 2 votes 1 answer 142 views ### Deriving the joint probability distribution of a transformed rv Suppose that A and B are 2 r.v with joint probability distribution given by f_{AB}(a,b) = -\frac{1}{2}(ln(a)+ln(b)), if 0 \lt a \lt 1 and 0 \lt b \lt 1, and 0 otherwise. Define X = A+B and Y =... 1 vote 1 answer 92 views ### Finding the probability of a bivariate joint random variable Let A=(M,N)^T be a bivariate random variable with joint density defined by$$f_{M,N}(m,n) = \frac{3}{2 \pi} \sqrt{m^2+n^2}$$if m^2+n^2<1 and 0 otherwise. Let B = (F,G)^T be given by$$\... 76 views

### Absolute value of expected value

Given $(A_1,A_2)$ to be the bivariate random variable in $\mathbb{R}^2$ with mean $0$ and cov($A_1,A_2)$ $= -1.05$ and $Var(Z_1) = Var(Z_2) = 1.05$. How do I find the expected value of $E(|A_1A_2|)$? ...
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### Finding the expected value of a bivariate gaussian distribution

Suppose that $(A,B)$ have a standard Gaussian distribution on $\mathbb{R}^2$. How do I find the expected value for $\mathbb{E}[max(3.9A+B,A+3.9B)]$? I know that A and B follow the standard normal ...
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### Proof that $\mathbb{E}\bigl[1-\Phi(x_1 - \theta_1) \mid \theta_2 \leq x_2-σx_1+σθ_1\bigr]$ is increasing in $\sigma$

Numerically it appears that the following function is increasing (or at least non-decreasing) in $σ$, $$f(x_1,x_2;σ)=∫_{-∞}^{∞}∫_{-∞}^{x_2-σx_1+σθ_1}[1-Φ(x_1-θ_1)]φ(θ_1)φ(θ_2)dθ_2dθ_1$$ where $\phi$ ...
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### Finding $P(X \leq a | Y = k)$ where $X\sim\operatorname{Exp}(\lambda)$ and $Y = [X]$ where $[\cdot]$ rounds up to the nearest integer.

$Y = [X]$ where $[\cdot]$ rounds up to the nearest integer. It's given that $X \sim \operatorname{Exp}(\lambda)$. The first part asks me to show that $Y \sim \operatorname{Geo}(1-e^{-\lambda})$. The ...
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### Guessing a function that maximises the variance?

I'm doing a question that discusses a six-sided fair die being rolled and then discussing the numbers that appear on the top and on the side facing you. The rest of the question discusses the ...
Let $$(X,Y)\sim N(\mu^\rightarrow ,\Sigma)$$ we cannot assume anything about the dependancy between X and Y. Can we assume the following? $$X\sim N(\mu_x,\sigma ^2_x)~,~Y\sim N(\mu_y, \sigma^2_y)$$...
Suppose (X, Y) is Bivariate Normal, with X,Y ~ N(0, 1), and Corr(X, Y) = $\rho$ , please find E(Y|X) and E(X|Y). I don't have a clue how to solve this, anyone could give me a pointer? Just for the ...